We study the problem of pattern formation on a lattice. A maximum-likelihood growth algorithm is formulated to supplement the integral solution of Laplace’s equation. This allows us to study diffusion-limited aggregation and viscous flow in porous media. Our deterministic solution for the aggregation process displays the same fractal dimension (D≊1.7) as Monte Carlo diffusion-limited aggregation. We also compare our solution to Laplace’s equation with the experimentally observed pattern in one limit of a Hele-Shaw cell experiment. Lastly, we present a second deterministic algorithm which results in a pattern with tips which appear parabolic.

A set of deterministic continuum equations is proposed to represent diffusion-limited aggregation. A solution using a Green’s-function method gives growth very similar to discrete simulations. The growth equations have no external noise. The relationship to viscous fingering and other growth processes is explored.

A number of new features of the overlapping Lorentz gas are obtained using a method based on describing the system as a random walk on a disordered lattice. The velocity correlation function is shown to have a contribution which decays like $t^{\mathrm{-}[2+1/(d\mathrm{-}1\left)\right]}$. This term is the dominant long-time tail for d>3 and above the percolation threshold. New values are obtained for the exponents describing the vanishing of the diffusion coefficient and the intermediate-time tail near threshold.

This work investigates the critical behavior of a micellar solution pentaethylene glycol-mono-n-dodecyl ether (${\mathrm{C}}_{12}$${\mathrm{E}}_5$)+water when the critical mixing point is approached within the two-phase region. The coexistence curve with exponent β≃(1/4) and light scattering measurements in the two coexisting phases are reported. The temperature dependence of the isothermal osmotic compressibility and the correlation length in the two coexisting phases have been found to obey power laws which disagree with those expected for the mean-field theory and also for the three-dimensional Ising-like model.

Relativistic quantum electrodynamic perturbation theory is applied to the high-Z few-electron system. Interaction of the bound electrons with the radiation field is treated as a perturbation that results in the electron-electron interaction and radiative corrections. The leading perturbations, of order α, are examined. For two-electron atoms, a precise evaluation of the one-exchanged-photon correction in the 1 ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$, 2 ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$, 2 ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$, and 2 ${}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}$ states is made.

The exact solution for the problem of a harmonic oscillator of frequency ${\omega }_0$ and mass $M_0$ ${\cos }^2$(νt) is extended to include the effect of a driving force $M_0$$f_0$ cos(λt+φ). With φ≠0, catastrophic resonances occur when λ=(${\omega }_0^2$+${\nu }^2$${)}^{1/2}$±ν. Pseudoperiodic states exist, provided that λ=${\omega }_0$ and λ,ν are commensurate. The corresponding quasienergies are finite in the cases φ=±π/2.

The growth of the Rayleigh-Schrödinger perturbation-theory (RSPT) energy coefficients for a one-dimensional harmonic oscillator perturbed by a potential ${\mathrm{gx}}^4$ is determined by the discontinuity of the energy, regarded as a function of complex g, across the negative g axis. That discontinuity has a perturbation expansion of the form g${\mathcal{’}}^{\mathrm{-}n\mathrm{-}1/2}$$e^{\mathrm{-}1/3g\mathcal{’}}$J$b_n^{\left(N\right)}$ (3g’${)}^N$, where g’=-g. The quasisemiclassical method for solving the Schrödinger equation by asymptotic expansion—which can be carried out term by term in closed form for the anharmonic oscillator to obtain both the dominant RSPT expansion and the subdominant $e^{\mathrm{-}1/3g\mathcal{’}}$ expansion—is developed here in detail. The notion of Borel summation is used to keep the role of the subdominant expansion unambiguous. Fifty terms in the $e^{\mathrm{-}1/3g\mathcal{’}}$ expansion—so-called Bender-Wu coefficients—are obtained for the first three states.

The dispersion of the collective excitation in the n=4 shells of xenon has been calculated, including all significant partial-wave contributions, by a time-dependent, self-consistent-field method. The calculation reveals details of the physical mechanism that produces the dispersion. The calculated dispersion coefficient is compared with experimental results.

We study the polarization dynamics due to the Breit-Rabi interaction of a muon in the 1S atomic state, with a nucleus of spin I having arbitrary initial polarization and orientation. We derive the equations of motion for the polarization in the commutator approach in two different bases for the density matrix. We find that the commonly used Cartesian basis is not very convenient for the present problem. In it, the 4(2I${+1)}^2$-1 polarization equations break only into two closed coupled subsets, the longitudinal and the transverse one, with a modest additional decoupling in the transverse sector, in presence of an external magnetic field. We have found an optimal basis in which these equations reduce to systems of four coupled differential equations, independent of the value of I, which can be readily solved for arbitrary boundary conditions. We determine the degree of muon repolarization, derive the equations for the longitudinal and transverse muon-spin-rotation spectroscopy, and point out the advantage of nuclear target polarizations in these applications.

A finite-basis method [B. R. Johnson and W. P. Reinhardt, Phys. Rev. A 28, 1930 (1983)] recently introduced for calculating bound-free transition amplitudes is extended here to apply to the Coulomb potential. This method is based upon the use of complex-coordinate techniques to calculate matrix elements of the time-independent wave operators of formal scattering theory, and avoids explicit enforcement of coordinate-space boundary conditions. Numerical experiments aiming at the calculation of the known hydrogenic ground-state photoionization amplitude have been made with use of different known forms of the two-body Coulomb wave operator. We have successfully reproduced the hydrogenic photoionization amplitude, but only within an accuracy and precision of about 1%. The most plausible explanation for these numerical difficulties is associated with the singular behavior of the ‘‘iɛ’’ limit in Coulomb scattering theory. In shorter-range problems, this limit is conveniently implemented by complex-coordinate techniques or complex-coordinate techniques in conjunction with Padé extrapolation. However, neither of these methods effectively implements the ɛ→0 limit in the Coulomb case.

By the Hellmann-Feynman theorem, the density n(r) of many electrons in the presence of external potential v(r) obeys the relationships F$d^3$r n(r)∇v(r)=0 and F$d^3$r n(r)r×∇v(r)=0. By the virial theorem, the interacting kinetic and electron-electron repulsion expectation values obey 2T[n]+$V_{\mathrm{ee}}$[n]=-F$d^3$r n(r)r⋅∇[δT/δn(r)+δ$V_{\mathrm{ee}}$/δn(r)]. The exchange energy functional $E_x$[n] and potential $v_x$([n];r)≡δ$E_x$/δn(r) must satisfy $E_x$[n]+F$d^3$r n(r)r⋅∇$v_x$([n];r)=0, while the correlation energy and potential must satisfy $E_c$[n]+F$d^3$r n(r)r⋅∇$v_c$([n];r)<0. Somewhat counterintuitively, it is not true that T[$n_{\gamma }$]=${\gamma }^2$T[n] and $V_{\mathrm{ee}}$[$n_{\gamma }$]=γ$V_{\mathrm{ee}}$[n], where $n_{\gamma }$(r)≡${\gamma }^3$n(γr) is a scaled density with scale factor γ≠1. In fact, it is impossible to partition the exact Hohenberg-Kohn functional into a piece that scales as ${\gamma }^2$ and a piece that scales as γ, even if complete freedom with the partitioning is allowed. Instead there are universal scaling inequalities.

For instance, T[$n_{\gamma }$]+$V_{\mathrm{ee}}$[$n_{\gamma }$]<${\gamma }^2$T[n]+ γ$V_{\mathrm{ee}}$[n] and T[$n_{\gamma }$]+γ$V_{\mathrm{ee}}$[$n_{\gamma }$]>${\gamma }^2$(T[n ]+$V_{\mathrm{ee}}$[n]), and consequent inequalities involving $E_c$[n]. All the above virial and scaling requisites are universal in that they are independent of external potential and they must hold for arbitrary proper n. In addition, for the ground-state energy (E) and n of any atom or molecule at its equilibrium nuclear configuration, there is the inequality E<-$T_s$[n], where $T_s$ is the noninteracting kinetic energy. In the closed-shell tight-binding limit, the correlation potential obeys F$d^3$r n(r)r⋅∇$v_c$([n];r)=0, and so cannot be a monotonic function of r for an atom in this limit.

Further, (∂/∂γ)$E_c$[$n_{\gamma }$]${\Vert }_{\gamma =1\mathrm{}}$=E ${}_c{}^{}{}_{}{}^{}$[n]+$T_c$[n]=-F$d^3$r n(r)r⋅∇$v_c$([n];r), which implies that the exact $E_c$ should be fairly insensitive to scaling. With the help of the ionization-potential theorem, it is argued that the exact $v_c$([n];r) in an atom often has a positive part. Common approximations to the correlation potential are examined for their effects upon the highest occupied Kohn-Sham orbital energy and the density moment 〈$r^2$〉, and these effects are found to be related. Further improvements needed in the approximate correlation potentials are relatively large, but not nearly so large as those recently suggested for the atoms Ne, Ar, Kr, and Xe: The discrepancy between theoretical values of 〈$r^2$〉 from Hartree-Fock or configuration-interaction calculations, and experimental values from measured diamagnetic susceptibilities, is tentatively resolved in favor of theory.

To calculate the nonzero dipole moment of isotopic molecules such as HD or the correct dissociation limit of ions such as ${\mathrm{HD}}^{+}$ via the usual approach requires cumbersome, expansive nonadiabatic corrections to the Born-Oppenheimer approximation. Such a complication is herein shown unnecessary; the ‘‘best adiabatic’’ approximation of Pack and Hirschfelder [J. Chem. Phys. 52, 521 (1970)] provides a single straightforward formalism which gives such effects correctly and yet retains the potential energy curve concept. Simple example calculations show the size of these symmetry-breaking effects to be at the level of high-resolution spectroscopy for ${\mathrm{HD}}^{+}$ but very large in ions such as dtμ, which contains a deuteron, a triton, and a negative muon.

A quantum-mechanical linear-response formalism is further developed to deal with the energy-level shifts of excited states of atoms near an arbitrary flat interface. We establish physical interpretations for different components of these energy-level shifts, and as examples treat atoms in free space, near a perfect conductor, near a metal, and near a dielectric waveguide. The effects of surface excitations such as surface plasmons and waveguide modes are considered in detail; we point out certain striking differences in the effects of these particular excitations.

A novel procedure is devised in order to obtain closed-form expressions of the Dirac-Coulomb radial $r^t$ integrals in terms of the Dirac energy ɛ={1+${\mathrm{Z}}^2$${\mathrm{\alpha }}^2$/[v+(${\mathrm{k}}^2$-${\mathrm{Z}}^2$ ${\mathrm{\alpha }}^2$${)}^{1/2}$${]}^2$${\mathit{\}}}^{\mathrm{-}1/2}$, where v=n-‖k‖, and of the Dirac quantum number k${=\left(\mathrm{-}1\right)}^{j+l+1/2\mathrm{}}$(j+(1/2)). In this procedure, well adapted for symbolic computation, the fundamental array of the $r^t$ radial integrals is obtained from the $r^{t\mathrm{-}1}$ array.

A nondegenerate perturbation theory is studied using the adiabatic-theorem formalism. A new factorization of the adiabatic time-evolution operator analogous to that obtained by Morita in his derivation of the linked-cluster theorem is given by carrying out a series of intermediate representations. This leads to a new expansion for the perturbed eigenvectors ‖α〉 and energies $E_{\alpha }$ which are analytically expressed with respect to two matrix series ${\mathrm{scrV}}^{\left(n\right)}$ and $E^{\left(n\right)}$. The nth term of each series (${\mathrm{scrV}}^{\left(n\right)}$ and $E^{\left(n\right)}$) includes perturbational contributions of power p, p≥n. An explicit recurrence, ${\mathrm{scrV}}^{\left(n\right)}$=V(${\mathrm{scrV}}^{n\mathrm{-}1)}$, $E^{\left(n\mathrm{-}1\right)}$), $E^{\left(n\right)}$=E(${\mathrm{scrV}}^{\left(n\mathrm{-}1\right)}$, $E^{\left(n\mathrm{-}1\right)}$) is derived and allows us to calculate these matrices to an arbitrary iteration order (n). This is a central feature of our approach since the eigenvectors and eigenvalues can be generated, to this arbitrary order, by a simple numerical program.

The three-center, two-electron molecular ion ${\mathrm{H}}_3$${\mathrm{}}^{+}$ has been studied using the channel-coupling array theory of n-particle scattering. The aim has been to determine how well the simplest approximations to the theory can handle electron correlation in a well-studied, three-center system. Two sets of approximate calculations of the ground-state energy of ${\mathrm{H}}_3$${\mathrm{}}^{+}$ are described. In the first, a channel-truncation approximation was used to yield a set of three coupled equations involving the three ${\mathrm{H}}_2$+${\mathrm{H}}^{+}$ channels. The channel-component wave functions were then approximated by hydrogenic 1s states; both the energy and equilibrium separation were poor. For the second set, a technique was developed in which the ${\mathrm{H}}_2$ cluster in each of the preceding ${\mathrm{H}}_2$+${\mathrm{H}}^{+}$ channels was subpartitioned into an H + H subchannel. The use of hydrogenic 1s states for the H + H subchannel components yielded much more accurate results: E(R=1.67$a_0$)=-1.307 a.u., compared to a valence-bond calculation with 1s states E(R=1.66$a_0$)=-1.298 a.u. As in previous calculations of this type, the results were obtained with relative computational ease.

We have measured the cross section σ(4F) for the collisional process Na(3P)+Na(3P)→Na(4F) +Na(3S). The experiment is based on the measurement of the intensities for 4F, 4D, and 3P fluorescence emissions in sodium vapor excited with a cw dye laser. σ(4F) is determined relative to σ(4D), which has been measured in an absolute way in previous experiments. Influence on σ(4F) and σ(4D) of the energy-transfer process Na(4F) +Na(3S)⇄Na(4D)+Na(3S) is also discussed. A theoretical evaluation of σ(4F) and σ(4D) relative to a chosen molecular channel is given and compared with the experimental determinations.

The bombarding energy dependence of the K-shell ionization probability $P_K$ was measured in elastic proton scattering on ${}_{}{}^{138}{}_{}{}^{}\mathrm{Ba}$ at energies near the 69-keV-wide $f_{7/2}$ isobaric-analog resonance at 9.965 MeV. The best simultaneous fit to $P_K$ in the scattering angle intervals 45°–80° and 100°–135° yields $R_0$=-0.30±0.30 for the tangent of the phase angle of the semiclassical monopole ionization amplitude.

L-shell x-ray production cross sections by ${}_1{}^1{}_{}{}^{}$${\mathrm{H}}^{+}$ ions are reported. The data are compared to the first Born approximation (plane-wave Born approximation for direct ionization and Oppenheimer-Brinkman-Kramers approximation for electron capture) and to the ECPSSR (energy-loss and Coulomb-deflection effects, perturbed stationary-state approximation with relativistic correction) theory. The energy of the protons ranged from 0.25 to 2.5 MeV in steps of 0.25 MeV. The targets used in these measurements were ${}_{28}{}^{}{}_{}{}^{}\mathrm{Ni}$, ${}_{29}{}^{}{}_{}{}^{}\mathrm{Cu}$, ${}_{32}{}^{}{}_{}{}^{}\mathrm{Ge}$, ${}_{33}{}^{}{}_{}{}^{}\mathrm{As}$, ${}_{37}{}^{}{}_{}{}^{}\mathrm{Rb}$, ${}_{38}{}^{}{}_{}{}^{}\mathrm{Sr}$, ${}_{39}{}^{}{}_{}{}^{}\mathrm{Y}$, ${}_{40}{}^{}{}_{}{}^{}\mathrm{Zr}$, and ${}_{46}{}^{}{}_{}{}^{}\mathrm{Pd}$. The first Born theory generally agrees with the data found in the literature at high energies and overpredicts them below 1.5 MeV. The ECPSSR predictions are in better agreement with experimental cross sections. At 0.25 MeV our data, however, are underestimated by this theory and tend to agree with the first Born approximation.

Effects of P-violating and CP-violating weak interactions in the heavy atoms Rb, Cs, Au, and Tl are calculated in a variety of potential models, including the relativistic Hartree-Fock model. Excitation energies and hyperfine constants are also calculated using these models and compared with experiment as a control. Values determined for the P-violating electric dipole matrix elements for Cs(6$s_{1/2}$→7$s_{1/2}$) and Tl(6$p_{1/2}$→7$p_{1/2}$) are in agreement with other similar calculations. The spread in values of the P- and CP-violating electric dipole matrix elements among the various models is found to be about 20%. A program is outlined for reducing this spread by systematically extending the present calculations to higher orders in the electron-electron interaction.